English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this note we consider a quantum mechanical particle moving inside aninfinitesimally thin layer constrained by a parabolic well in the $x$-directionand, moreover, in the presence of an impurity modelled by an attractiveGaussian potential. We investigate the Birman-Schwinger operator associated toa model assuming the presence of a Gaussian impurity inside the layer and provethat such an integral operator is Hilbert-Schmidt, which allows the use of themodified Fredholm determinant in order to compute the bound states created bythe impurity. Furthermore, we consider the case where the Gaussian potentialdegenerates to a $\delta$-potential in the $x$-direction and a Gaussianpotential in the $y$-direction. We construct the corresponding self-adjointHamiltonian and prove that it is the limit in the norm resolvent sense of asequence of corresponding Hamiltonians with suitably scaled Gaussianpotentials. Satisfactory bounds on the ground state energies of allHamiltonians involved are exhibited.

frontiers in physics

The Birman-Schwinger Operator for a Parabolic Quantum Well in a  Zero-Thickness Layer in the Presence of a Two-Dimensional Attractive Gaussian  Impurity